Axiom A

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In mathematics, Axiom A or Smale's Axiom A systems define certain types of dynamical systems that are particularly chaotic. The term originates with Stephen Smale.

[edit] Definition

Let M be a smooth manifold. We say that a diffeomorphism f: M\to M satisfies (Smale's) Axiom A (or that f is an Axiom A diffeomorphism) if

  1. The nonwandering set Ω(f) has a hyperbolic structure;
  2. The set of periodic points of f is dense in Ω(f), so that the closure is the non-wandering set itself: \overline{\operatorname{Per}(f)} = \Omega(f).

Sometimes, Axiom A diffeomorphisms are called hyperbolic diffeomorphisms, because the portion of M where the "interesting" dynamics occur (namely, Ω(f)) has a hyperbolic behaviour.

This article incorporates material from Axiom A on PlanetMath, which is licensed under the GFDL.

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